Remark 2.7 
Note that if \(\omega =0\)  and \(\lambda >0\)  then the epidemic ultimately spreads to all individuals in the components of the underlying graph \(G(n, \mu /n)\)  that have initial infectives,
so the distribution of the final size \(T^{(n)}\)  is independent of \(\lambda \) . In particular, for \(\mu >1\)  and \(\lambda >\mu ^{-1}\)  (so \(R_0>1\) ), \(\rho (\mu )=\tau _{SI}(\mu , \lambda , 0, 0)\)  gives the fraction of individuals in the giant component of \(G(n, \mu /n)\)  in the limit as \(n \rightarrow \infty \) . Setting \(\omega =0\)  in Theorem REF (b) yields that \(\rho (\mu )\)  is the unique solution in \((0,1)\)  of \(1-x={\rm e}^{-\mu x}\)  (see, for example, [1]}, Theorem 2.3.2).

